Nematic liquid crystals in a spatially step-wise magnetic field

We study the molecular reorientation induced by a textured external field in a nematic liquid crystal (nLC). In particular, we consider an infinitely wide cell with strong planar anchoring boundary conditions, subjected to a spatially periodic piecewise magnetic field. In the framework of the Frank's continuum theory, we use the perturbation analysis to study in detail the field-induced splay-bend Fréedericksz transition. A numerical approach, based on the finite differences method, is instead employed to solve the fully nonlinear equations. At high field strengths, an analytic approach allows us to draw the bulk profile of the director in terms of elliptic integrals. Finally, through the application of the Bruggeman texture hydrodynamics theory, we qualitatively discuss on the LCs piecewise director configuration under sliding interfaces, which can be adopted to actively regulate friction. Our study opens the pathway for the application of highly controlled nLC texturing for tribotronics.